A212581 -id:A212581 - cp's OEIS frontend

1, 1, 2, 5, 20, 102, 626, 4458, 36144, 328794, 3316944, 36755520, 443828184, 5800823880, 81591320880, 1228888215960, 19733475278880, 336551479543440, 6075437671458000, 115733952138747600, 2320138519554562560, 48827468196234035280, 1076310620915575933440

Offset: 0

Tom Roby, May 21 2012

nonn

Also the number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> bac where a

Also the number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> cba where a

Also the number of permutations of [n] avoiding consecutive triples j, j+1, j-1. a(4) = 20 = 4! - 4 counts all permutations of [4] except 1342, 2314, 3421, 4231. - Alois P. Heinz, Apr 14 2021

			From _Alois P. Heinz_, May 22 2012: (Start)
a(3) = 5: {123, 132}, {213}, {231}, {312}, {321}.
a(4) = 20: {1234, 1243, 1324}, {1342}, {1423}, {1432}, {2134}, {2143}, {2314}, {2341, 2431}, {2413}, {3124}, {3142}, {3214}, {3241}, {3412}, {3421}, {4123, 4132}, {4213}, {4231}, {4312}, {4321}. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..450
Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
S. Linton, J. Propp, T. Roby, and J. West, Equivalence classes of permutations under various relations generated by constrained transpositions, 2011 arXiv:1111.3920 [math.CO], J. Int. Seq. 15 (2012) #12.9.1

Cf. A174072, A210667, A210668, A210669, A210671, A212417, A212581.

Column k=0 of A343535.

b:= proc(s, x, y) option remember; `if`(s={}, 1, add(
      `if`(x=0 or x-y<>1 or j-x<>1, b(s minus {j}, y, j), 0), j=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Apr 14 2021
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [1$2, 2, 5, 20][n+1],
       n*a(n-1)+3*a(n-2)-(2*n-2)*a(n-3)+(n-2)*a(n-5))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 14 2021

a[n_] := a[n] = If[n < 5, {1, 1, 2, 5, 20}[[n+1]],
     n*a[n-1] + 3*a[n-2] - (2n - 2)*a[n-3] + (n-2)*a[n-5]];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 20 2022, after Alois P. Heinz *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*(1-x^2))^k)) \\ Seiichi Manyama, Feb 20 2024

From Seiichi Manyama, Feb 20 2024: (Start)
G.f.: Sum_{k>=0} k! * ( x * (1-x^2) )^k.
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * (n-2*k)! * binomial(n-2*k,k). (End)

a(9)-a(22) from Alois P. Heinz, Apr 14 2021

A212432 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> cba where a

Original entry on oeis.org

1, 1, 2, 4, 16, 84, 536, 3912, 32256, 297072, 3026112, 33798720, 410826624, 5399704320, 76317546240, 1154312486400, 18604815528960, 318348065548800, 5763746405053440, 110086912964367360, 2212209395234979840, 46657233031296706560, 1030510550216174469120

Offset: 0

Tom Roby, Jun 21 2012

nonn

Also number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> bac <--> cba where a < b < c.

			From _Alois P. Heinz_, Jun 22 2012: (Start)
a(3) = 4: {123, 132, 321}, {213}, {231}, {312}.
a(4) = 16: {1234, 1243, 1324, 1432, 3214}, {1342}, {1423}, {2134}, {2143}, {2314}, {2341, 2431, 4123, 4132, 4321}, {2413}, {3124}, {3142}, {3241}, {3412}, {3421}, {4213}, {4231}, {4312}.
a(5) = 84: {12345, 12354, 12435, 12543, 13245, 13254, 14325, 32145, 32154}, {12453}, ..., {53421}, {54213}, {54231}.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..450
Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
S. Linton, J. Propp, T. Roby, and J. West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, arXiv:1111.3920, 2011 [math.CO], J. Int. Seq. 15 (2012) #12.9.1

Cf. A212580, A212581.

From Seiichi Manyama, Feb 21 2024: (Start)
G.f.: Sum_{k>=0} k! * ( x * (1-2*x^2) )^k.
a(n) = Sum_{k=0..floor(n/3)} (-2)^k * (n-2*k)! * binomial(n-2*k,k). (End)

a(9) from Alois P. Heinz, Jun 23 2012

a(10)-a(22) from Alois P. Heinz, Apr 14 2021

A212433 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac <--> cba, where a

Original entry on oeis.org

1, 1, 2, 3, 13, 71, 470, 3497, 29203, 271500, 2786711, 31322803, 382794114, 5054810585, 71735226535, 1088920362030, 17607174571553, 302143065676513, 5484510055766118, 104999034898520903, 2114467256458136473, 44682676397748896010, 988663144904696100347

Offset: 0

Tom Roby, Jun 21 2012

nonn

			From _Alois P. Heinz_, Jun 23 2012: (Start)
a(3) = 3: {123, 132, 213, 321}, {231}, {312}.
a(4) = 13: {1234, 1243, 1324, 1432, 2134, 3214}, {1342}, {1423}, {2143}, {2314}, {2341, 2431, 3241, 4123, 4132, 4213, 4321}, {2413}, {3124}, {3142}, {3412}, {3421}, {4231}, {4312}.
a(5) = 71: {12345, 12354, 12435, 12543, 13245, 13254, 14325, 21345, 21354, 21435, 21543, 32145, 32154}, {12453}, ..., {53412}, {53421}, {54231}.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..450
Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
S. Linton, J. Propp, T. Roby, and J. West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, arXiv:1111.3920, 2011 [math.CO]

Cf. A212432, A212580, A212581.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x*((1-x^2)^2/(1-x^3)-x^2))^k)) \\ Seiichi Manyama, Feb 25 2024

G.f.: Sum_{k>=0} k! * ( x * ((1-x^2)^2/(1-x^3) - x^2) )^k. - Seiichi Manyama, Feb 25 2024

a(8)-a(9) from Alois P. Heinz, Jun 23 2012

a(10)-a(22) from Alois P. Heinz, Apr 14 2021

Showing 1-3 of 3 results.

cp's OEIS Frontend

A212580 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb where a

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A212432 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> cba where a

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A212433 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac <--> cba, where a

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions